CN107526105A - A kind of wave-field simulation staggering mesh finite-difference method - Google Patents

A kind of wave-field simulation staggering mesh finite-difference method Download PDF

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CN107526105A
CN107526105A CN201710676672.8A CN201710676672A CN107526105A CN 107526105 A CN107526105 A CN 107526105A CN 201710676672 A CN201710676672 A CN 201710676672A CN 107526105 A CN107526105 A CN 107526105A
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张懿洁
高静怀
彭济根
柳清伙
张金淼
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Xiamen University
Xian Jiaotong University
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Xiamen University
Xian Jiaotong University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V1/00—Seismology; Seismic or acoustic prospecting or detecting
    • G01V1/28—Processing seismic data, e.g. analysis, for interpretation, for correction
    • G01V1/30—Analysis
    • G01V1/303—Analysis for determining velocity profiles or travel times
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V1/00—Seismology; Seismic or acoustic prospecting or detecting
    • G01V1/28—Processing seismic data, e.g. analysis, for interpretation, for correction
    • G01V1/30—Analysis
    • G01V1/301—Analysis for determining seismic cross-sections or geostructures
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V2210/00—Details of seismic processing or analysis
    • G01V2210/60—Analysis
    • G01V2210/62—Physical property of subsurface
    • G01V2210/622—Velocity, density or impedance
    • G01V2210/6222—Velocity; travel time
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V2210/00—Details of seismic processing or analysis
    • G01V2210/60—Analysis
    • G01V2210/66—Subsurface modeling
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V2210/00—Details of seismic processing or analysis
    • G01V2210/60—Analysis
    • G01V2210/67—Wave propagation modeling
    • G01V2210/673—Finite-element; Finite-difference

Abstract

The invention discloses a kind of wave-field simulation staggering mesh finite-difference method, this method is applied to the wave-field simulation of the wave equation of any medium.The present invention is directed in high-order staggering mesh finite-difference method because of wild effect caused by difference coefficient solution, is provided the Algorithm for Solving difference coefficient based on generalized circular matrix and is effectively overcome this problem.This method can provide precise and stable wave-field simulation result；The technical scheme is easily achieved, applicability is extensive.

Description

A kind of wave-field simulation staggering mesh finite-difference method
Technical field
The invention belongs to seismic exploration technique field, is related to method for numerical simulation, especially a kind of precise and stable wave field Simulate staggering mesh finite-difference method.
Background technology
Wavefield forward modeling technology is the important foundation of seismic wave detection, and recognizes the crucial work of seismic reservoir ripple response Tool, the precondition of parametric inversion and imaging even more in seismic data interpretation.Because finite difference method realization is easy, can be flexible Complex dielectrics is handled, has been widely used in seismic forward modeling simulation.Finite difference method is by time in wave equation and sky Between derivative represented with the form of difference coefficient, it is achieved thereby that time and space is discrete.Finite difference method has many kinds, including aobvious Formula and finite differential method, time-domain and frequency domain finite difference method, staggered-mesh and rotationally staggered grid finite difference Divide method etc..
Staggering mesh finite-difference method refers in the departure process of wave equation, medium parameter and equation variable point It is not placed on different mesh points, so as to improve the accuracy of method.In order to further improve staggering mesh finite-difference method Accuracy to meet actual demand, high-order finite difference method method can be used, or reduce time step or space lattice size.This Outside, optimization method, such as simulated annealing method, least square method etc. can also be used to calculate the difference of finite difference method Coefficient has obtained more preferable accuracy.But high-order finite difference method method is most convenient.
However, when difference order is sufficiently high, it can find that finite difference method is no longer stable by dispersion analysis.Reason is The coefficient matrix in difference coefficient formula is calculated close to unusual, but still is solved using the method for matrix inversion.At present, do not send out The method for now solving the problem.
The content of the invention
The shortcomings that it is an object of the invention to overcome above-mentioned prior art, there is provided a kind of precise and stable wave-field simulation is interlocked Grid finite difference method, it is directed in high-order staggering mesh finite-difference method because unstable existing caused by difference coefficient solution As providing the Algorithm for Solving difference coefficient based on generalized circular matrix and effectively overcoming this problem.This method can provide accurate steady Fixed wave-field simulation result；The technical scheme is easily achieved, applicability is extensive.
The purpose of the present invention is achieved through the following technical solutions：
This wave-field simulation staggering mesh finite-difference method, numerical value is carried out using high-order staggering mesh finite-difference method During simulation, the wild effect as caused by matrix inversion technique solves difference coefficient, asked using the algorithm based on generalized circular matrix Difference coefficient is solved, obtains stable and accurate wave field result.
Further, above wave-field simulation staggering mesh finite-difference method, specifically includes following steps：
1) staggering mesh finite-difference method and Taylor method of deploying are based on, 2M rank precision is carried out to first order spatial derivative Expansion, by contrasting the coefficient of equal sign both ends variable, obtains the difference coefficient a of 2M rank precision space derivationsmSolution formula：
2) system of linear equations in the formula (1) in step 1) is carried out changing member：
So, the coefficient matrix in formula (2) is generalized circular matrix, is written as
Wherein, xm=(2m-1)2。
Further, above step 2) in, it is the difference coefficient a in solution formula (2)m, using following algorithm：
First, double circulation is carried out, outer circulation is that k is incremented to M-1 from 1, and interior circulation is decremented to k+1 from M for n, operates b (n)=b (n)-x (k) × b (n-1)；
Then, then double circulation is carried out, outer circulation is that k is decremented to 1 from M-1, and circulation is that n is incremented to from k+1 in first M, b (n)=b (n)/(x (n)-x (n-k)) is operated, circulation is that n is incremented to M-1, operation b (n)=b (n)-b (n from k in second +1)；
Symbol is expressed as x in algorithmM×1, wherein x (m)=(2m-1)2, bM×1=[1 0 ... 0]；By this algorithm process, Final b (n) is exactly the variable of system of linear equations in formula (2).
Further, above step 1) in, in staggering mesh finite-difference method, the following institute of 2M rank precision space derivations Show：
In above formula, h is space lattice size, am(m=1,2 ..., M) is the difference system of staggering mesh finite-difference method Number；
It is rightWithTaylor expansion is carried out, and contrasts equation left and right ends letter The coefficient of number f n order derivatives, obtains the difference coefficient a of 2M rank space derivationsmThe formula (1) of solution formula, i.e. step 1).
The invention has the advantages that：
The wave-field simulation staggering mesh finite-difference method of the present invention is applied to the wave field mould of the wave equation of any medium Intend, it is directed in high-order staggering mesh finite-difference method because of wild effect caused by difference coefficient solution, is provided based on model The Algorithm for Solving difference coefficient that moral covers matrix effectively overcomes this problem.
Further, the present invention is convenient, flexible, suitable for the forward simulation of any medium wave equation, including acoustics, elasticity, Viscoplasticity, poroelasticity, anisotropic medium.
Further, the present invention can effectively overcome wild effect caused by matrix inversion, can calculate the difference of accurate stable Coefficient, so as to obtain more stable accurately wave field information.
Brief description of the drawings
Fig. 1 is the dispersion curve of different difference orders, wherein τ=1ms, h=10m,V=2000m/s, β=kh, Difference coefficient is tried to achieve by the method for matrix inversion；
Fig. 2 is the dispersion curve of different difference orders, wherein τ=1ms, h=10m,V=2000m/s, β=kh, Difference coefficient is tried to achieve by the method based on generalized circular matrix；
Fig. 3 is the even resilient medium velocity z-component calculated by (a) matrix inversion technique and (b) present invention 0.2s's Wave field snapshot；
Fig. 4 is the medium parameter schematic diagram of actual reservoir geophysical model；
Fig. 5 is z points of the speed of the actual reservoir geophysical model calculated by (a) matrix inversion technique and (b) present invention The earthquake record of amount.
Embodiment
When the present invention is directed to using high-order staggering mesh finite-difference method progress numerical simulation, asked by matrix inversion technique Wild effect caused by solving difference coefficient, solves this using the Algorithm for Solving difference coefficient based on generalized circular matrix and asks Topic, it can obtain more stable and accurate wave field result.
The present invention is described in detail with example below in conjunction with the accompanying drawings.
1) the difference coefficient a of 2M ranks space derivationmSolution formula
In staggering mesh finite-difference method, 2M rank precision space derivations are as follows
Here, h is space lattice size, am(m=1,2 ..., M) is the difference system of staggering mesh finite-difference method Number.
It is rightWithTaylor expansion is carried out, and contrasts equation left and right ends letter The coefficient of number f n order derivatives, can obtain the difference coefficient a of 2M rank space derivationsmSolution formula：
Based on formula (5), difference coefficient a is asked for using the method for matrix inversionm, when Fig. 1 gives different difference orders Dispersion curve, frequency dispersion parameter is as follows：
Wherein,
Obviously, when δ is equal to 1, without numerical solidification, when δ is more than or less than 1, there is numerical solidification.Can from figure Go out, the precision of 30 rank staggering mesh finite-difference methods is less than 8 rank staggering mesh finite-difference methods.Obviously, this and " difference rank Number it is higher, precision is higher " the fact be not inconsistent.
When table 1 provides different M, the conditional number of coefficient matrix in formula (5)：
During 1 different M of table, the conditional number of coefficient matrix in formula (5)
Obviously, the conditional number of the matrix sharply increases with M increase, and when M is 10, conditional number is sufficiently large and matrix Close to unusual.Therefore, when asking for the difference coefficient of high-order staggering mesh finite-difference method using the method for matrix inversion, gesture Wild effect must be produced.
2) difference coefficient asks for the conversion of formula
Handled by algebraically, formula (5) can be changed to：
So, the coefficient matrix of above formula is changed into generalized circular matrix：
Wherein, xm=(2m-1)2。
3) algorithm of difference coefficient is asked for based on generalized circular matrix
To solve equation Vx=b, can be realized using the algorithm in table 2.This algorithm can be also expressed as
First, double circulation is carried out, outer circulation is that k is incremented to M-1 from 1, and interior circulation is decremented to k+1 from M for n, operates b (n)=b (n)-x (k) × b (n-1)；
Then, then double circulation is carried out, outer circulation is that k is decremented to 1 from M-1, and circulation is that n is incremented to from k+1 in first M, b (n)=b (n)/(x (n)-x (n-k)) is operated, circulation is that n is incremented to M-1, operation b (n)=b (n)-b (n from k in second +1)。
Symbol is expressed as x in algorithmM×1, wherein x (m)=(2m-1)2, bM×1=[10 ... 0].By the iteration of the algorithm, The b finally given is exactly required unknown quantity x.
The equation Vx=b of table 2 solution, wherein V are generalized circular matrix
Fig. 2 gives the dispersion curve that the algorithm based on generalized circular matrix calculates, it can be seen that difference order is higher, numerical value frequency Scattered smaller, i.e., method is more stable accurate.
Stability analysis
In order to verify the stability of the present invention, by whether meeting stability condition explanation.
Under two-dimensional case, stability condition expression formula is as follows
Wherein, τ is time step, vmaxFor the maximal rate of model.For sound wave medium, it is SVEL；Elasticity is situated between Matter, it is p wave interval velocity；Poroelasticity medium, it is fast p wave interval velocity.
Table 3 gives whether the present invention meets aforementioned stable condition, wherein
Stability condition in the case of 3 different difference orders of table, wherein τ=1ms, h=10m, v=2000m/s
Numerical result
Apply the present invention to the numerical simulation of even resilient medium and actual reservoir geophysical model, it is effective to verify Property.
Even resilient medium parameter：P wave interval velocity 4000m/s, S wave velocity 2000m/s, density 2600kg/m3.Numerical simulation When, time step 1ms, space lattice size is 10m, and model size is [0,2000m] × [0,2000m].Focus is dominant frequency 30Hz, time delay 1/30s Ricker wavelets, positioned at model center.Difference order is 36, and difference coefficient is respectively by matrix inversion Method and the inventive method are tried to achieve.The wave field snapshot of speed z-component is as shown in Figure 3.It can be seen that, the number of results of matrix inversion technique Value frequency dispersion is very serious, and the result that the present invention calculates does not have numerical solidification.
The P ripples of actual reservoir geophysical model, S wave velocities and density are as shown in Figure 4.Change model from top to bottom to distinguish For：Loose sand, mud stone, tight sand, mud stone, tight sand, mud stone.Inverted trapezoidal target is included wherein in third layer medium Area, include gas-bearing formation, oil reservoir and water layer.During numerical simulation, time step 1ms, space lattice size is 10m, and model size is [0,58km]×[0,12.8km].Focus is dominant frequency 10Hz, time delay 0.1s Ricker wavelets, positioned at [29km, 10m].Difference Exponent number is 36, and difference coefficient is tried to achieve by the method and the inventive method of matrix inversion respectively.The earthquake record of speed z-component is as schemed Shown in 5.It can be seen that, the result value frequency dispersion of matrix inversion technique is very serious, and the result that the present invention calculates does not have numerical value frequency Dissipate.Illustrate the difference coefficient more accurate stable that the present invention calculates.

Claims (4)

  1. A kind of 1. wave-field simulation staggering mesh finite-difference method, it is characterised in that using high-order staggering mesh finite-difference side When method carries out numerical simulation, the wild effect as caused by matrix inversion technique solves difference coefficient, using based on vandermonde square The Algorithm for Solving difference coefficient of battle array, obtains stable and accurate wave field result.
  2. 2. wave-field simulation staggering mesh finite-difference method according to claim 1, it is characterised in that
    1) staggering mesh finite-difference method and Taylor method of deploying are based on, 2M rank precision exhibitions are carried out to first order spatial derivative Open, by contrasting the coefficient of equal sign both ends variable, obtain the difference coefficient a of 2M rank precision space derivationsmSolution formula：
    <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msup> <mn>1</mn> <mn>1</mn> </msup> </mtd> <mtd> <msup> <mn>3</mn> <mn>1</mn> </msup> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>1</mn> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mn>1</mn> <mn>3</mn> </msup> </mtd> <mtd> <msup> <mn>3</mn> <mn>3</mn> </msup> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mn>1</mn> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mn>3</mn> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>a</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mi>M</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    2) system of linear equations in the formula (1) in step 1) is carried out changing member：
    <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mn>1</mn> <mn>2</mn> </msup> </mtd> <mtd> <msup> <mn>3</mn> <mn>2</mn> </msup> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mn>1</mn> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mn>3</mn> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>a</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>3</mn> <msub> <mi>a</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> <msub> <mi>a</mi> <mi>M</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>1</mi> </mtd> </mtr> <mtr> <mtd> <mi>0</mi> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    So, the coefficient matrix in formula (2) is generalized circular matrix, is written as
    <mrow> <mi>V</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mn>1</mn> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mn>2</mn> </msub> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <msub> <mi>x</mi> <mi>M</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <msup> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <msup> <msub> <mi>x</mi> <mi>M</mi> </msub> <mrow> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, xm=(2m-1)2。
  3. 3. wave-field simulation staggering mesh finite-difference method according to claim 2, it is characterised in that in step 2), be Difference coefficient a in solution formula (2)m, using following algorithm：
    First, double circulation is carried out, outer circulation is that k is incremented to M-1 from 1, and interior circulation is decremented to k+1, operation b (n)=b for n from M (n)-x(k)×b(n-1)；
    Then, then double circulation is carried out, outer circulation is that k is decremented to 1 from M-1, and circulation is that n is incremented to M from k+1 in first, is grasped Make b (n)=b (n)/(x (n)-x (n-k)), circulation is that n is incremented to M-1 from k in second, operates b (n)=b (n)-b (n+1)；
    Symbol is expressed as x in algorithmM×1, wherein x (m)=(2m-1)2, bM×1=[10 ... 0]；By this algorithm process, final b (n) be exactly system of linear equations in formula (2) variable.
  4. 4. wave-field simulation staggering mesh finite-difference method according to claim 2, it is characterised in that in step 1), In staggering mesh finite-difference method, 2M rank precision space derivations are as follows：
    <mrow> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>f</mi> </mrow> <mrow> <mo>&amp;part;</mo> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <msub> <mi>a</mi> <mi>m</mi> </msub> <mo>{</mo> <mi>f</mi> <mo>&amp;lsqb;</mo> <mi>x</mi> <mo>+</mo> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mn>2</mn> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>-</mo> <mi>f</mi> <mo>&amp;lsqb;</mo> <mi>x</mi> <mo>-</mo> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mn>2</mn> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>}</mo> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <msup> <mi>h</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
    In above formula, h is space lattice size, am(m=1,2 ..., M) is the difference coefficient of staggering mesh finite-difference method；
    It is rightWithTaylor expansion is carried out, and contrasts equation left and right ends function f's The coefficient of n order derivatives, obtain the difference coefficient a of 2M rank space derivationsmThe formula (1) of solution formula, i.e. step 1).
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